Hello! Welcome to this wiki. It is time to review rational expressions.Please complete the questions below using this wiki page. Place your answers under the discussion tab at the top of this page.
1. What value of may not be used for k with inverse variation?
2. What variable is varying jointly in the joint variation definition? _
3. What equation is given for direct variation in the definition?
4. What is the answer to 9.1 Practice Problem #2?
5. What is the name of the graph of a rational expression in 9.2? _
6. What is the domain and range for 9.2 Practice Problem #6? _
7. In 9.3, what are the asymptotes and x intercepts of #7? _
8. In 9.5, what is the answer to #14? _
9. In 9.6, what is stated as a real-life example to use solving rational equations? _
10. What was the video about?
Mary Sweitzer - Notes
Cheyenne Stambaugh - Notes
Alex Schmitt - Multi-Media
Devin Taylor - Real Life
Gabriella Sallada- notes, editor
Amanda Saravia- Practice Problems 9.1 Inverse and Joint Variation Definitions: *Inverse Variation – y=k/x ; where k cannot be 0
*Constant of variation – nonzero constant k; y varies inversely with x
Joint Variation – a quantity varies directly as the product of two or more other quantities; z=kxy where z cannot be 0
Direct Variation – y = kx
Real Life- Finding the speed of a whirlpool's current.
Practice Problems
Tell whether x and y are direct variation or inverse variation or neither. 1. xy = 8
2. x = y/3 The variables x and y vary inversely. Use the given values to write and equation relating x and y. 3. x = 10, y = 2 The variables z varies jointly with x and y. Use the given values to find an equation that relates the variables. 4. x = 4, y = 3, z = 24
9.2 Graphing Simple Rational Functions Definitions:
*Rational Function – function of the form f(x) = p(x)/q(x)
*Hyperbola – graph of a rational function
- x- axis is a horizontal asymptote
- y- axis is a vertical asymptote
- The domain and range are all nonzero real numbers
*Branches – two symmetrical parts of a hyperbola
- For each point (x,y) there is a corresponding point (-x,-y) on the other branch All rational functions of the form y = (a/ x – h) + k will have asymptotes at y = k and x = h. To draw this graph, choose some x values, plug them into the equation and plot them. Finally connect the dots and draw in the asymptotes
Real Life- Finding the frequency of an approaching ambulance siren.
Practice Problems Graph the function. State the domain and range. 5. y = -2/x
6. y = 2/(x-3) + 1
9.3 Real Life- Finding the energy expenditure of a parakeet. Practice Problems State the asymptotes and x intercepts. 7. y = x/x^2 + 4
8. y = x^2 + 2 / x^2 - x - 6 9.4 Multiplying and Dividing Rational Expressions When simplifying rational expressions, the following property can be used:
[Let a, b, and c be nonzero real numbers or varible expressions.]
ac/bc= a/b <=== The common factor of c can be divided out
The same rules that apply to multiplying numerical fractions applies when multiplying rational expressions. The numerators are multiplied, along with the denominators, and then just written as a new fraction in simplified form.
a/b x c/d ---> a x c / b x d = ac/bd <== this can even be simplified, if possible.
When dividing rational expressions, just take the first expression multiplied by the recipricol of the second expression.
a/b / c/d ----> a/b x d/c = ad/bc <==Simplify, if possible
Real Life- Finding the average number of acres per farm. Practice Problems Simplify the rational expression, if possible. 9. y^2 - 81 / 2y - 18
10. x + 3 / x^2 + 6x + 9 Multiply the rational expressions and then simplify.
11. x^2 + 2x - 3 / x + 2 times x^2 + 2x / x^2 -1
12. 12 - x / 3 times 3 / x - 12 Divide the rational expressions and then simplify. 13. 48x^2 / y divided by 36xy^2 / 5
14. x^2 / x^2 - 1 divided by 3x / x + 1 9.5 Addition, Subtraction and Complex Fractions
How to solve the addition and subtraction of complex fractions depends on whether or not the expressions have like or unlike denominators. If they have like denominators then simply add or subtract the numerators and then put the result over the common denominator.
If they have unlike denominators, then you need to find the LCD (Least Common Denominator). To do this, just multiply the denominators of each expression by each other as if they were one. (i.e multiply by x+3/ x+3 instead of just x+3) Then rewrite the expression and simplify.
Complex Fractions Complex fractions are formed of two fractional expressions, one on top of the other. There are two methods for simplifying complex fractions. The first method is fairly obvious: find common denominators for the complex numerator and complex denominator, convert the complex numerator and complex denominator to their respective common denominators, combine everything in the complex numerator and in the complex denominator into single fractions, and then, once you've got one fraction (in the complex numerator) divided by another fraction (in the complex denominator), you flip-n-multiply. (Remember that, when you are dividing by a fraction, you flip the fraction and turn the division into multiplication.)
This method looks like this:
4+ 1/x_ = (4x/x + 1/x) = (4x+1/x) 3+ 2/ x^2 (3x^2/x^2 + 2/x^2) (3x^2+2/x^2)
= (4x+1/x)(x^2/3x^2+2) <-- an x can be taken out of both fractions to simplify.
*Multiply*
x(4x+1)
_4x^2+x3x^2+2 3x^2+2
Everything cancels at this point, so this is the final answer.
4x^2+x /3x^2+ 2
Real Life- Modeling the total number of male college graduates. Practice Problems Find the LCD. 15. 3 / x + 4 x / x^2 - 16 x + 2 / 4
16. 13 / x^2 - 2x + 1 4 / x^2 - 1 5 / x (x + 1) Perform the indicated operations and then simplify. 17. 2x - 1 / x^2 - x - 2 minus 1 / x - 2
18. 2x / x + 2 minus 8 / x^2 + 2x plus 3 / x Simplify the complex fraction.
19. (1 / x + 9 plus 1 / 5) / ( 2 / x^2 + 10x + 9) 9.6 Solving Rational Equations__
To solve a rational equation, multiply each term on both sides of the equation by the LCD of the terms. Simplify and solve the resulting polynomial equation.
To solve a rational equation for which each side of the equation is a single rational expression, use cross multiplying. When adding and subtracting rational expressions, you had to find a common denominator. Now that you have equations, you are allowed to multiply through (because you have two sides to multiply on) and get rid of the denominators entirely. In other words, you still need to find the common denominator, but you don't necessarily need to use it in the same way.
Example: Solve the following equation: Least Common Denominator is 3x.
7/x - 1/3x = 5/3 -----> Write the original equation.
3x (7/x - 1/3x) = 3x(5/3) ------> Multiply each side by the LCD
21 - 1 = 5x -----> Simplify
20 = 5x ----->Subtract
4=x ------> Divide each side by 5
Practice Problems Solve by using LCD or cross multiplying. Check each solution. 20. 3x/x-2 = 1 + 6/x-2
21. 6+5x/3x = 7/x
22. 6/x - 7x/5 = x/10
9.6 Real Life- Finding the year which a certain amount of rodeo prize money was earned.
Answers to Practice Problems 1. inverse
2. direct
3. y = 20/x
4. z = 2xy 5. curve in the 2nd quadrant with x = 0 and y = 0 as asymptotes, another curve in 4th quadrant with x = 0 and y = 0 as asymptotes
Domain- x= all real but 0 Range- y = all real but 0
6. curve in first quadrant and 3rd going into fourth quadrant with x = 3 and y = 1 as an asymptotes
Domain- x = all real but 3 Range- y = all real but 1 7. Asymptotes- x = 2 and x = -2 and y = 0
X-Intercepts - (0,0)
8. Asymptotes- x = 3 and x = -2 and y = 1
X-Intercepts- no intercepts 9. y + 9 / 2
10. 1 / x + 3
11. x (x + 3) / x + 1
12. -1
13. 20x / 3y^3
14. x / 3 (x - 1) 15. 4 (x + 4) ( x - 4)
16. x (x + 1) (x - 1) (x - 1)
17. 1 / x + 1
18. 2x - 1 / x
19.( x + 14) (x + 1) / 10 20. no solution
21. x = 3
22. x = 2 and x = -2
Rational Function Graph
Hello! Welcome to this wiki. It is time to review rational expressions.Please complete the questions below using this wiki page. Place your answers under the discussion tab at the top of this page.
1. What value of may not be used for k with inverse variation?
2. What variable is varying jointly in the joint variation definition? _
3. What equation is given for direct variation in the definition?
4. What is the answer to 9.1 Practice Problem #2?
5. What is the name of the graph of a rational expression in 9.2? _
6. What is the domain and range for 9.2 Practice Problem #6? _
7. In 9.3, what are the asymptotes and x intercepts of #7? _
8. In 9.5, what is the answer to #14? _
9. In 9.6, what is stated as a real-life example to use solving rational equations? _
10. What was the video about?
Mary Sweitzer - Notes
Cheyenne Stambaugh - Notes
Alex Schmitt - Multi-Media
Devin Taylor - Real Life
Gabriella Sallada- notes, editor
Amanda Saravia- Practice Problems
9.1 Inverse and Joint Variation
Definitions:
*Inverse Variation – y=k/x ; where k cannot be 0
*Constant of variation – nonzero constant k; y varies inversely with x
- Joint Variation – a quantity varies directly as the product of two or more other quantities; z=kxy where z cannot be 0
- Direct Variation – y = kx
Real Life- Finding the speed of a whirlpool's current.Practice Problems
Tell whether x and y are direct variation or inverse variation or neither.
1. xy = 8
2. x = y/3
The variables x and y vary inversely. Use the given values to write and equation relating x and y.
3. x = 10, y = 2
The variables z varies jointly with x and y. Use the given values to find an equation that relates the variables.
4. x = 4, y = 3, z = 24
9.2 Graphing Simple Rational Functions
Definitions:
*Rational Function – function of the form f(x) = p(x)/q(x)
*Hyperbola – graph of a rational function
- x- axis is a horizontal asymptote
- y- axis is a vertical asymptote
- The domain and range are all nonzero real numbers
*Branches – two symmetrical parts of a hyperbola
- For each point (x,y) there is a corresponding point (-x,-y) on the other branch
All rational functions of the form y = (a/ x – h) + k will have asymptotes at y = k and x = h. To draw this graph, choose some x values, plug them into the equation and plot them. Finally connect the dots and draw in the asymptotes
Real Life- Finding the frequency of an approaching ambulance siren.
Practice Problems
Graph the function. State the domain and range.
5. y = -2/x
6. y = 2/(x-3) + 1
9.3 Real Life- Finding the energy expenditure of a parakeet.
Practice Problems
State the asymptotes and x intercepts.
7. y = x/x^2 + 4
8. y = x^2 + 2 / x^2 - x - 6
9.4 Multiplying and Dividing Rational Expressions
When simplifying rational expressions, the following property can be used:
[Let a, b, and c be nonzero real numbers or varible expressions.]
ac/bc= a/b <=== The common factor of c can be divided out
The same rules that apply to multiplying numerical fractions applies when multiplying rational expressions. The numerators are multiplied, along with the denominators, and then just written as a new fraction in simplified form.
a/b x c/d ---> a x c / b x d = ac/bd <== this can even be simplified, if possible.
When dividing rational expressions, just take the first expression multiplied by the recipricol of the second expression.
a/b / c/d ----> a/b x d/c = ad/bc <==Simplify, if possible
Real Life- Finding the average number of acres per farm.
Practice Problems
Simplify the rational expression, if possible.
9. y^2 - 81 / 2y - 18
10. x + 3 / x^2 + 6x + 9
Multiply the rational expressions and then simplify.
11. x^2 + 2x - 3 / x + 2 times x^2 + 2x / x^2 -1
12. 12 - x / 3 times 3 / x - 12
Divide the rational expressions and then simplify.
13. 48x^2 / y divided by 36xy^2 / 5
14. x^2 / x^2 - 1 divided by 3x / x + 1
9.5 Addition, Subtraction and Complex Fractions
How to solve the addition and subtraction of complex fractions depends on whether or not the expressions have like or unlike denominators. If they have like denominators then simply add or subtract the numerators and then put the result over the common denominator.
If they have unlike denominators, then you need to find the LCD (Least Common Denominator). To do this, just multiply the denominators of each expression by each other as if they were one. (i.e multiply by x+3/ x+3 instead of just x+3) Then rewrite the expression and simplify.
Complex Fractions
Complex fractions are formed of two fractional expressions, one on top of the other. There are two methods for simplifying complex fractions. The first method is fairly obvious: find common denominators for the complex numerator and complex denominator, convert the complex numerator and complex denominator to their respective common denominators, combine everything in the complex numerator and in the complex denominator into single fractions, and then, once you've got one fraction (in the complex numerator) divided by another fraction (in the complex denominator), you flip-n-multiply. (Remember that, when you are dividing by a fraction, you flip the fraction and turn the division into multiplication.)
3+ 2/ x^2 (3x^2/x^2 + 2/x^2) (3x^2+2/x^2)
= (4x+1/x)(x^2/3x^2+2) <-- an x can be taken out of both fractions to simplify.
*Multiply*
x(4x+1)
_4x^2+x3x^2+2 3x^2+24x^2+x /3x^2+ 2
Real Life- Modeling the total number of male college graduates.
Practice Problems
Find the LCD.
15. 3 / x + 4 x / x^2 - 16 x + 2 / 4
16. 13 / x^2 - 2x + 1 4 / x^2 - 1 5 / x (x + 1)
Perform the indicated operations and then simplify.
17. 2x - 1 / x^2 - x - 2 minus 1 / x - 2
18. 2x / x + 2 minus 8 / x^2 + 2x plus 3 / x
Simplify the complex fraction.
19. (1 / x + 9 plus 1 / 5) / ( 2 / x^2 + 10x + 9)
9.6 Solving Rational Equations__
To solve a rational equation, multiply each term on both sides of the equation by the LCD of the terms. Simplify and solve the resulting polynomial equation.
To solve a rational equation for which each side of the equation is a single rational expression, use cross multiplying.
When adding and subtracting rational expressions, you had to find a common denominator. Now that you have equations, you are allowed to multiply through (because you have two sides to multiply on) and get rid of the denominators entirely. In other words, you still need to find the common denominator, but you don't necessarily need to use it in the same way.
Example:
Solve the following equation:
Least Common Denominator is 3x.
7/x - 1/3x = 5/3 -----> Write the original equation.
3x (7/x - 1/3x) = 3x(5/3) ------> Multiply each side by the LCD
21 - 1 = 5x -----> Simplify
20 = 5x ----->Subtract
4=x ------> Divide each side by 5
Practice Problems
Solve by using LCD or cross multiplying. Check each solution.
20. 3x/x-2 = 1 + 6/x-2
21. 6+5x/3x = 7/x
22. 6/x - 7x/5 = x/10
9.6 Real Life- Finding the year which a certain amount of rodeo prize money was earned.
Answers to Practice Problems
1. inverse
2. direct
3. y = 20/x
4. z = 2xy
5. curve in the 2nd quadrant with x = 0 and y = 0 as asymptotes, another curve in 4th quadrant with x = 0 and y = 0 as asymptotes
Domain- x= all real but 0 Range- y = all real but 0
6. curve in first quadrant and 3rd going into fourth quadrant with x = 3 and y = 1 as an asymptotes
Domain- x = all real but 3 Range- y = all real but 1
7. Asymptotes- x = 2 and x = -2 and y = 0
X-Intercepts - (0,0)
8. Asymptotes- x = 3 and x = -2 and y = 1
X-Intercepts- no intercepts
9. y + 9 / 2
10. 1 / x + 3
11. x (x + 3) / x + 1
12. -1
13. 20x / 3y^3
14. x / 3 (x - 1)
15. 4 (x + 4) ( x - 4)
16. x (x + 1) (x - 1) (x - 1)
17. 1 / x + 1
18. 2x - 1 / x
19.( x + 14) (x + 1) / 10
20. no solution
21. x = 3
22. x = 2 and x = -2